.TH std::atanh(std::complex) 3 "2024.06.10" "http://cppreference.com" "C++ Standard Libary"
.SH NAME
std::atanh(std::complex) \- std::atanh(std::complex)

.SH Synopsis
   Defined in header <complex>
   template< class T >                       \fI(since C++11)\fP
   complex<T> atanh( const complex<T>& z );

   Computes the complex arc hyperbolic tangent of z with branch cuts outside the
   interval [−1; +1] along the real axis.

.SH Parameters

   z - complex value

.SH Return value

   If no errors occur, the complex arc hyperbolic tangent of z is returned, in the
   range of a half-strip mathematically unbounded along the real axis and in the
   interval [−iπ/2; +iπ/2] along the imaginary axis.

   Error handling and special values

   Errors are reported consistent with math_errhandling.

   If the implementation supports IEEE floating-point arithmetic,

     * std::atanh(std::conj(z)) == std::conj(std::atanh(z))
     * std::atanh(-z) == -std::atanh(z)
     * If z is (+0,+0), the result is (+0,+0)
     * If z is (+0,NaN), the result is (+0,NaN)
     * If z is (+1,+0), the result is (+∞,+0) and FE_DIVBYZERO is raised
     * If z is (x,+∞) (for any finite positive x), the result is (+0,π/2)
     * If z is (x,NaN) (for any finite nonzero x), the result is (NaN,NaN) and
       FE_INVALID may be raised
     * If z is (+∞,y) (for any finite positive y), the result is (+0,π/2)
     * If z is (+∞,+∞), the result is (+0,π/2)
     * If z is (+∞,NaN), the result is (+0,NaN)
     * If z is (NaN,y) (for any finite y), the result is (NaN,NaN) and FE_INVALID may
       be raised
     * If z is (NaN,+∞), the result is (±0,π/2) (the sign of the real part is
       unspecified)
     * If z is (NaN,NaN), the result is (NaN,NaN)

.SH Notes

   Although the C++ standard names this function "complex arc hyperbolic tangent", the
   inverse functions of the hyperbolic functions are the area functions. Their argument
   is the area of a hyperbolic sector, not an arc. The correct name is "complex inverse
   hyperbolic tangent", and, less common, "complex area hyperbolic tangent".

   Inverse hyperbolic tangent is a multivalued function and requires a branch cut on
   the complex plane. The branch cut is conventionally placed at the line segments
   (-∞,-1] and [+1,+∞) of the real axis.

   The mathematical definition of the principal value of the inverse hyperbolic tangent
   is atanh z =

   ln(1+z) - ln(1-z)
   2

   .
   For any z, atanh(z) =

   atan(iz)
   i

   .

.SH Example


// Run this code

 #include <complex>
 #include <iostream>

 int main()
 {
     std::cout << std::fixed;
     std::complex<double> z1(2.0, 0.0);
     std::cout << "atanh" << z1 << " = " << std::atanh(z1) << '\\n';

     std::complex<double> z2(2.0, -0.0);
     std::cout << "atanh" << z2 << " (the other side of the cut) = "
               << std::atanh(z2) << '\\n';

     // for any z, atanh(z) = atanh(iz) / i
     std::complex<double> z3(1.0, 2.0);
     std::complex<double> i(0.0, 1.0);
     std::cout << "atanh" << z3 << " = " << std::atanh(z3) << '\\n'
               << "atan" << z3 * i << " / i = " << std::atan(z3 * i) / i << '\\n';
 }

.SH Output:

 atanh(2.000000,0.000000) = (0.549306,1.570796)
 atanh(2.000000,-0.000000) (the other side of the cut) = (0.549306,-1.570796)
 atanh(1.000000,2.000000) = (0.173287,1.178097)
 atan(-2.000000,1.000000) / i = (0.173287,1.178097)

.SH See also

   asinh(std::complex) computes area hyperbolic sine of a complex number
   \fI(C++11)\fP             (\\({\\small\\operatorname{arsinh}{z}}\\)arsinh(z))
                       \fI(function template)\fP
   acosh(std::complex) computes area hyperbolic cosine of a complex number
   \fI(C++11)\fP             (\\({\\small\\operatorname{arcosh}{z}}\\)arcosh(z))
                       \fI(function template)\fP
                       computes hyperbolic tangent of a complex number
   tanh(std::complex)  (\\({\\small\\tanh{z}}\\)tanh(z))
                       \fI(function template)\fP
   atanh
   atanhf              computes the inverse hyperbolic tangent
   atanhl              (\\({\\small\\operatorname{artanh}{x}}\\)artanh(x))
   \fI(C++11)\fP             \fI(function)\fP
   \fI(C++11)\fP
   \fI(C++11)\fP
   C documentation for
   catanh
